[Paper Review] Quantizing the geodesic flow via adapted complex structures

William D. Kirwin|arXiv (Cornell University)|Aug 7, 2014

Geometry and complex manifolds12 references3 citations

TL;DR

This paper resolves convergence issues in geometric quantization of the geodesic flow by introducing a Wick-rotated dragging projection using adapted complex structures. It rigorously computes the quantum Hamiltonian as $-\frac{\hbar^2}{2}(\Delta - \frac{1}{6}S)$, confirming the $c=1/6$ ordering constant via complex-time evolution and Gaussian integrals in normal coordinates.

ABSTRACT

The geometric quantization of the geodesic flow on a compact Riemannian manifold via the BKS "dragging projection" yields the Laplacian plus a scalar curvature term. To avoid convergence issues, the standard construction involves somewhat unnatural hypotheses that do not hold in typical examples. In this paper, we use adapted complex structures to make sense of a Wick-rotated version of the dragging projection which avoids the convergence issues.

Motivation & Objective

• To resolve convergence problems in the standard BKS dragging projection construction for geometric quantization of the geodesic flow on compact Riemannian manifolds.
• To provide a mathematically rigorous alternative to the divergent integrals and asymptotic approximations in the classical dragging projection method.
• To establish the quantum Hamiltonian as $-\frac{\hbar^2}{2}(\Delta - \frac{1}{6}S)$ using a Wick-rotated version of the dragging projection.
• To employ adapted complex structures and complex-time geodesic flows to define a well-behaved BKS pairing without regularization.
• To confirm the $c=1/6$ value for the scalar curvature term in the quantum Hamiltonian under anti-Kohn-Nirenberg ordering.

Proposed method

• Introduce a Wick rotation by analytically continuing time $t$ to $it$, transforming the real geodesic flow into a complex-time flow that avoids divergence.
• Utilize adapted complex structures of Guillemin–Stenzel/Lempert–Szőke to define the time-$i$ geodesic flow as the pushforward of the vertical tangent bundle via rescaling in $T^*M$.
• Define the Wick-rotated dragging projection via the BKS pairing between vertically constant sections and sections evolved under the imaginary-time geodesic flow.
• Compute the BKS pairing using Taylor expansions of the volume form and metric in normal coordinates, up to second order in fiber coordinates $p$.
• Approximate the resulting integrals by extending the domain from a ball to $\mathbb{R}^n$ and evaluate Gaussian integrals using standard formulas.
• Derive the quantum Hamiltonian by taking the derivative at $t=0$ of the evolved state, yielding the operator $-\frac{\hbar^2}{2}(\Delta - \frac{1}{6}S)$.

Experimental results

Research questions

• RQ1 How can the divergent integrals in the standard dragging projection be rigorously resolved in geometric quantization of the geodesic flow?
• RQ2 What is the role of adapted complex structures in enabling a well-defined complex-time evolution for the geodesic flow?
• RQ3 Why does the Wick-rotated dragging projection yield the $c=1/6$ constant in the quantum Hamiltonian $-\frac{\hbar^2}{2}(\Delta - cS)$?
• RQ4 How does the BKS pairing behave under analytic continuation to imaginary time, and what ensures its convergence?
• RQ5 Can the scalar curvature term in the quantum Hamiltonian be derived without regularization or asymptotic expansions?

Key findings

• The Wick-rotated dragging projection provides a convergent, rigorous alternative to the standard dragging projection, eliminating the need for regularization.
• The quantum Hamiltonian for the geodesic flow is rigorously derived as $-\frac{\hbar^2}{2}(\Delta - \frac{1}{6}S)$, confirming the $c=1/6$ value for anti-Kohn-Nirenberg ordering.
• The BKS pairing in the Wick-rotated framework is well-defined and finite, with the volume form's Taylor expansion in normal coordinates yielding the $\frac{1}{12}p^j p^k R_{jk}$ correction term.
• Gaussian integral approximations over $\mathbb{R}^n$ accurately capture the leading-order behavior of the BKS pairing, with error terms $O(t^2)$.
• The imaginary-time geodesic flow corresponds to the pushforward of the vertical tangent bundle under the adapted complex structure, ensuring transversality and smoothness.
• The final result matches known physics results from path integral quantization, such as DeWitt’s $c=1/6$ and Bastianelli–Van Nieuwenhuizen’s derivation in the Bargmann-Fock representation.